Question

1. explain the difference between the following descriptions.

• four is less than x.
• four less than x

1. for parts (a)–(c), consider the numbers 0, 3, 5, 12, and 40.

a. which of the numbers are solutions to ( x + 6 > 20 )?
b. which of the numbers are solutions to ( 8.5 < 20 - w )?
c. which of the numbers are solutions to ( \frac{1}{2}n > 1 )

for problems 9 and 10, use the following information. it takes lisa 10 minutes to run 1 mile. lisa wants to run at a constant rate for more than 30 minutes.

1. which inequality can be used to determine the number of miles ( m ) that lisa wants to run at this rate?

a. ( m > 10 )
b. ( m > 30 )
c. ( 10m > 30 )
d. ( 30m > 10 )

1. which statement correctly describes the number of miles that lisa wants to run at this rate?

a. lisa wants to run at most 3 miles.
b. lisa wants to run more than 3 miles.
c. lisa wants to run at most 10 miles.
d. lisa wants to run more than 10 miles.

Explanation:

Step1: Solve Q7: Translate phrases to math

1. "Four is less than $x$": $4 < x$
2. "Four less than $x$": $x - 4$

Step2: Solve Q8a: Isolate $x$

$x + 6 > 20$
$x > 20 - 6$
$x > 14$
Test numbers: 0,3,5,12 <14; 40>14

Step3: Solve Q8b: Isolate $w$

$8.5 < 20 - w$
$w < 20 - 8.5$
$w < 11.5$
Test numbers: 0,3,5 <11.5; 12,40>11.5

Step4: Solve Q8c: Isolate $n$

$\frac{1}{2}n > 1$
$n > 1 \times 2$
$n > 2$
Test numbers: 0<2; 3,5,12,40>2

Step5: Solve Q9: Set up time inequality

Total time = $10m$, time >30: $10m > 30$

Step6: Solve Q10: Solve Q9's inequality

$10m > 30$
$m > \frac{30}{10}$
$m > 3$

Answer:

1. "Four is less than $x$" translates to the inequality $4 < x$. "Four less than $x$" translates to the algebraic expression $x - 4$.

8.
a. 40
b. 0, 3, 5
c. 3, 5, 12, 40

1. C. $10m > 30$
2. B. Lisa wants to run more than 3 miles.